Graphs, networks and design

This module is about using ideas from discrete mathematics to model problems, and representing these ideas through diagrams. The word ‘graphs’ refers to diagrams consisting of points joined by lines. These points may correspond to chemical atoms, towns, electrical terminals or anything that can be connected in pairs. The lines may be chemical bonds, roads, wires or other connections. The main topics of mathematical interest are graphs and digraphs; network flows; block designs; geometry; codes; and mathematical modelling. Application areas covered include communications; structures and mechanisms; electrical networks; transport systems; social networks; and computer science. To study this module you should have a sound knowledge of relevant mathematics provided by the appropriate OU level 2 study.

Modules count towards OU qualifications

OU qualifications are modular in structure; the credits from this undergraduate-level module could count towards a certificate of higher education, diploma of higher education, foundation degree or honours degree.

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What you will study

What codes are used by spacecraft in communicating with Earth? Where do you brace a framework to make it rigid? How many colours are needed for a map to ensure that neighbouring countries have different colours? How can you assign people to jobs for which they are qualified? These are some of the questions to be answered in the module. The problems range from those arising in technology, operational research and the sciences to puzzles of a recreational nature. We show the connections between problems in widely differing areas and describe methods for their solution that use the properties they have in common.

The material is presented in a down-to-earth manner, with the emphasis on solving problems and applying algorithms, rather than on abstract ideas and proofs.

The module is divided into three related areas: graphs, networks and design. The Introduction introduces two themes of the module, combinatorics and mathematical modelling, and illustrates them with examples from the three areas.

Graphs 1: Graphs and digraphs discusses graphs and digraphs in general, and describes the use of graph theory in genetics, ecology and music, and of digraphs in the social sciences. We discuss Eulerian and Hamiltonian graphs and related problems; one of these is the well-known Königsberg bridges problem.

Networks 1: Network flows is concerned with the problem of finding the maximum amount of a commodity (gas, water, passengers) that can pass between two points of a network in a given time. We give an algorithm for solving this problem, and discuss important variations that frequently arise in practice.

Design 1: Geometric design, concerned with geometric configurations, discusses two-dimensional patterns such as tiling patterns, and the construction and properties of regular and semi-regular tilings, and of polyominoes and polyhedra.

Graphs 2: Trees Trees are graphs occurring in areas such as branching processes, decision procedures and the representation of molecules. After discussing their mathematical properties we look at their applications, such as the minimum connector problem and the travelling salesman problem.

Networks 2: Optimal paths How does an engineering manager plan a complex project encompassing many activities? This application of graph theory is called ‘critical path planning’. It is one of the class of problems in which the shortest or longest paths in a graph or digraph must be found.

Design 2: Kinematic design The mechanical design of table lamps, robot manipulators, car suspension systems, space-frame structures and other artefacts depends on the interconnection of systems of rigid bodies. This unit discusses the contribution of combinatorial ideas to this area of engineering design.

Graphs 3: Planarity and colouring When can a graph be drawn in the plane without crossings? Is it possible to colour the countries of any map with just four colours so that neighbouring countries have different colours? These are two of several apparently unrelated problems considered in this unit.

Networks 3: Assignment and transportation If there are ten applicants for ten jobs and each is suitable for only a few jobs, is it possible to fill all the jobs? If a manufacturer supplies several warehouses with a product made in several factories, how can the warehouses be supplied at the least cost? These problems of the system-management type are answered in this unit.

Design 3: Design of codes Redundant information in a communication system can be used to overcome problems of imperfect reception. This section discusses the properties of certain codes that arise in practice, in particular cyclic codes and Hamming codes, and some codes used in space probes.

Graphs 4: Graphs and computing describes some important uses of graphs in computer science, such as depth-first and breadth-first search, quad trees, and the knapsack and travelling salesman problems.

Networks 4: Physical networks Graph theory provides a unifying method for studying the current through an electrical network or water flow through pipes. This unit describes the graphical representation of such networks.

Design 4: Block designs If an agricultural research station wants to test different varieties of a crop, how can a field be designed to minimise bias due to variations in the soil? The answer lies in block designs. This unit explains the concepts of balanced and resolvable designs and gives methods for constructing block designs.

Conclusion In this unit, many of the ideas and problems encountered in the module are reviewed, showing how they can be generalised and extended, and the progress made in finding solutions is discussed.

You will learn

Successful study of this module should enhance your skills in finding solutions to problems, interpreting mathematical results in real-world terms and communicating mathematical ideas clearly to both experts and non-experts.

Professional recognition

This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.

Teaching and assessment

Support from your tutor

You will have a tutor who will help you with the study material and mark and comment on your written work, and whom you can ask for advice and guidance. We may also be able to offer group tutorials or day schools that you are encouraged, but not obliged, to attend. Where your tutorials are held will depend on the distribution of students taking the module.

Contact us if you want to know more about study with The Open University before you register.

Assessment

The assessment details for this module can be found in the facts box above.

Please note that TMAs for all undergraduate mathematics and statistics modules must be submitted on paper as – due to technical reasons – we are unable to accept TMAs via our eTMA system.

Future availability

The details given here are for the module that starts in October 2015. We expect it to be available once a year.

Regulations

As a student of The Open University, you should be aware of the content of the Module Regulations and the Student Regulations which are available on our Essential documents website.

Course work includes:

4 Tutor-marked assignments (TMAs)

4 Computer-marked assignments (CMAs)

Examination

No residential school

Course satisfaction survey

Entry

This interfaculty module is intended for students with a variety of backgrounds. The more mathematically inclined will see how their mathematics can be used to solve problems, while those with a technological interest will learn to appreciate the use of a mathematical framework to relate different ideas.

This is an OU level 3 module. OU level 3 modules build on study skills and subject knowledge acquired from studies at levels 1 and 2. They are intended only for students who have recent experience of higher education in a related subject, preferably with The Open University.

A suitable preparation is a background in mathematics (such as Essential mathematics 1 (MST124) and Essential mathematics 2 (MST125) or their predecessors MST121 and MS221) and 60 credits at OU level 2 in mathematics, science or technology. In particular, familiarity with matrix multiplication and the matrix formulation of simultaneous equations would be an advantage, although these are reviewed in the module. Please note that you are more likely to complete this module successfully if you have acquired your prerequisite knowledge through passing these modules.

There is a diagnostic quiz to help you to determine whether you are adequately prepared for this module. Your regional or national centre may also be able to tell you where you can see reference copies of the suggested modules, or you can buy selected materials from Open University Worldwide Ltd.

Don’t worry if you haven’t studied technology before. If you rely on common sense and accept certain statements of a scientific or technological nature, you should have no difficulty. Whatever your experience you should find plenty to interest you, as long as you go along with the interdisciplinary nature of the module.

If you have any doubt about the suitability of the module, please speak to an adviser.

Register

Start

End

Fee

Register

03 Oct 2015

Jun 2016

Not yet available

Registration opens on 12/03/15

This module is expected to start for the last time in October 2018.

Ways to pay for this module

Open University Student Budget Account

The Open University Student Budget Accounts Ltd (OUSBA) offers a convenient 'pay as you go' option to pay your OU fees, which is a secure, quick and easy way to pay. Please note that The Open University works exclusively with OUSBA and is not able to offer you credit facilities from any other provider. All credit is subject to status and proof that you can afford the repayments.

You pay the OU through OUSBA in one of the following ways:

Register now, pay later – OUSBA pays your module fee direct to the OU. You then repay OUSBA interest-free and in full just before your module starts. 0% APR representative. This option could give you the extra time you may need to secure the funding to repay OUSBA.

Pay by instalments – OUSBA calculates your annual fees and spreads them out over up to a year, enabling you to pay your fees monthly and walk away with a qualification without any further debt. APR 5.1% representative.

Employer sponsorship

Studying with The Open University can boost your employability. OU qualifications are recognised and respected by employers for their excellence and the commitment they take to achieve one. They also value the skills that students learn and can apply in the workplace.

More than one in 10 OU students are sponsored by their employer, and over 30,000 employers have used the OU to develop staff so far. If the qualification you’ve chosen is geared towards your job or developing your career, you could approach your employer to see if they will sponsor you by paying some or all of the fees.

Your employer just needs to complete a simple form to confirm how much they will be paying and we will invoice them.

You won’t need to get your employer to complete the form until after you’ve chosen your modules.

Credit/debit card

You can pay part or all of your tuition fees upfront with a debit or credit card when you register for each module.

Gift vouchers

You can pay for part or all of your tuition fees with OU gift vouchers. Vouchers are currently available in the following denominations, £10, £20, £50 and £100.

Mixed payments

We know that sometimes you may want to combine payment options. You may, for example wish to pay part of your tuition fee with a debit card and pay the remainder in instalments through an Open University Student Budget Accounts (OUSBA).

For more information about combining payment options, speak to an adviser or request a call back.

Note: Your permanent address/domicile will affect your fee status and therefore the fees you are charged and any financial support available to you. The fee and funding information provided here is valid for courses starting before 31 July 2015. Fees normally increase annually in line with inflation. Fees for courses starting from 1 August 2015 will be available in March 2015.

This information was provided on 03/03/2015.

What's included

Module books, CDs, DVDs, software and a website.

You will need

CD player and DVD player (or computer able to play DVDs).

You require access to the internet at least once a week during the module to download module resources and assignments, submit assignments and to keep up to date with module news.

Computing requirements

You will need a computer with internet access to study this module as the study materials and activities are accessible via a web browser. You may also be required to perform other tasks, such as word processing, using spreadsheets, taking part in online forums, and submitting files to the university for assessment. The additional software for these tasks will either be provided or is freely available. For this module you will also need to install software provided by the OU on a disk or USB stick.

A Windows desktop or laptop computer running Windows 7 or later operating system is suitable for this module. You will be required to install Microsoft Windows specific software.

A netbook, tablet, smartphone or Linux computer that supports one of the browsers listed below may be suitable. The screen size should be at least 1024 (H) x 768 (W) pixels. If you intend to use one of these devices please ensure you have access to a suitable desktop or laptop computer in case you are unable to carry out all the module activities on your mobile device.

We recommend a minimum 1 Mbps internet connection and any of the following browsers:

Internet Explorer 9 and above

Apple Safari 7 and above

Google Chrome 31 and above

Mozilla Firefox 31 and above.

Note: using the latest version for your browser will maximise security when accessing the internet. Using company or library computers may prevent you accessing some internet materials or installing additional software.

If you have a disability

The many diagrams in the text and the computing element could be demanding if you have impaired sight or certain types of colourblindness. Written transcripts are available for the audio-visual material.

To use the module software you will need to spend considerable amounts of time using a personal computer although its use is not essential. It is designed to enhance the student’s learning experience but it is possible to pass the module without using it.

If you have particular study requirements please tell us as soon as possible, as some of our support services may take several weeks to arrange. Find out more about our services for disabled students.